Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{8x}{6x(2x + 5)} \div \dfrac{-2}{9(2x + 5)} $
Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{8x}{6x(2x + 5)} \times \dfrac{9(2x + 5)}{-2} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 8x \times 9(2x + 5) } { 6x(2x + 5) \times -2 } $ $ p = \dfrac{72x(2x + 5)}{-12x(2x + 5)} $ We can cancel the $2x + 5$ so long as $2x + 5 \neq 0$ Therefore $x \neq -\dfrac{5}{2}$ $p = \dfrac{72x \cancel{(2x + 5})}{-12x \cancel{(2x + 5)}} = -\dfrac{72x}{12x} = -6 $